The first part of this book is devoted to function spaces in Euclidean n-space with dominating mixed smoothness. Some new properties are derived and applied in the second part where weighted spaces with dominating mixed smoothness in arbitrary bounded domains in Euclidean n
-space are introduced and studied. This includes wavelet frames, numerical integration and discrepancy, measuring the deviation of sets of points from uniformity.
These notes are addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of Besov–Sobolev type. In particular, it will be of interest for researchers dealing with approximation theory, numerical integration and discrepancy.
Keywords: Function spaces, dominating mixed smoothness, spaces on domains, wavelets, Faber frames, Haar frames
Author(s): Hans Triebel
Series: EMS Series of Lectures in Mathematics
Publisher: European Mathematical Society
Year: 2019
Language: English
Pages: 212
Preface......Page 6
Basic notation and isotropic spaces......Page 10
Spaces with dominating mixed smoothness......Page 13
Atoms......Page 17
Wavelets......Page 21
Complements......Page 25
Distinguished spaces......Page 30
Homogeneity......Page 37
Non-smooth atoms......Page 40
Pointwise multipliers and localizations......Page 44
Pointwise multipliers: General assertions......Page 49
Local embeddings and isomorphic structure......Page 54
Intermezzo: Key problems......Page 62
Fourier multipliers......Page 63
Embeddings......Page 64
Traces......Page 66
Dichotomy......Page 71
Fatou property......Page 73
Extensions......Page 74
Diffeomorphisms......Page 75
Pointwise multipliers: Special assertions......Page 78
Multiplication algebras......Page 86
Pointwise multipliers, revisited......Page 92
Hölder inequalities......Page 96
Caloric wavelets and smoothing......Page 103
Thermic characterizations......Page 106
Tempered homogeneous spaces with negative smoothness......Page 111
Thermic characterizations, revisited......Page 117
Tempered homogeneous spaces with positive smoothness......Page 121
Tempered homogeneous spaces with general smoothness......Page 128
Introduction......Page 132
Definitions and basic properties......Page 134
Wavelet frames......Page 136
Preliminaries......Page 143
Sobolev spaces......Page 144
Besov spaces......Page 147
Motivations and preliminaries......Page 150
Spaces with boundary data......Page 159
Faber frames......Page 165
Haar frames......Page 175
Further comments and some embeddings......Page 180
Numerical integration: An example......Page 182
Discrepancy......Page 188
Bibliography......Page 194
Symbols......Page 206
Index......Page 210
